Time-space tradeoffs for set operations

This paper considers time-space tradeoffs for various set operations. Denoting the time requirement of an algorithm by T and its space requirement by S, it is shown that TS=Ω(n²) for set complementation and TS=Ω(n ^{3 2}) for set intersection, in the R-way branching program model. In the more restricted model of comparison branching programs, the paper provides two additional types of results. A tradeoff of TS=Ω(n^2-ε(n)), derived from Yao's lower bound for element distinctness, is shown for set disjointness, set union and set intersection [where ε(n)=O((logn)^{- 1 2})]. A bound of TS=Ω(n ^{3 2}) is shown for deciding set equality and set inclusion. Finally, a classification of set operations is presented, and it is shown that all problems of a large naturally arising class are as hard as the problems bounded in this paper.